TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 353, Number 8, Pages 3347–3370 S 0002-9947(01)02795-7 Article electronically published on April 9, 2001

WANDERING VECTOR MULTIPLIERS FOR UNITARY GROUPS

DEGUANG HAN AND D. LARSON

Abstract. A wandering vector multiplier is a which maps the set of wandering vectors for a unitary system into itself. A special case of unitary system is a discrete unitary group. We prove that for many (and perhaps all) discrete unitary groups, the set of wandering vector multipliers is itself a group. We completely characterize the wandering vector multipliers for abelian and ICC unitary groups. Some characterizations of special wandering vector multipliers are obtained for other cases. In particular, there are simple characterizations for diagonal and permutation wandering vector multipliers. Similar results remain valid for irrational rotation unitary systems. We also obtain some results concerning the wandering vector multipliers for those uni- tary systems which are the ordered products of two unitary groups. There are applications to wavelet systems.

0. Introduction The close relationship between standard wavelet theory and wandering vector theory for unitary systems has been systematically studied in [DL]. For both theo- retical and practical reasons, the most interesting unitary systems are those which are ordered products of two (or more) unitary groups. The study of this kind of unitary system is necessarily based on the study of the unitary groups. Also, the case of a “group unitary system” has independent interest, and some open problems remain. In this paper we will focus on the situation where the underlying is separable and the unitary system is countable. We will investigate the structure of the set of wandering vector multipliers for the group case, and for the ordered product of two groups. The latter includes wavelet systems and irrational rotation unitary systems. This article represents the final product of a project of the authors that was begun in the Spring of 1996. Let H be a separable Hilbert space. A unitary system U is a set of unitary operators acting on H which contains the identity operator I in B(H). If G1 and G2 are unitary groups, by the ordered product of G1 and G2, we will mean the unitary system U = {g1g2 : gi ∈Gi,i=1, 2}.Awandering subspace for U is a closed linear subspace M of H with the property that UM ⊥ VM for any different U and V in U. If in addition H = span UM,wecallM a complete wandering subspace for U. In the case that M = Cx for some unit vector x,wecallx a (complete) wandering vector for U if M is a (complete) wandering subspace for U. The set of all complete

Received by the editors February 5, 1998. 2000 Mathematics Subject Classification. Primary 46L10, 46L51, 42C40. Key words and phrases. Unitary system, wandering vector, wandering vector multiplier, , wavelet system. (DH) Participant, Workshop in Linear Analysis and Probability, Texas A&M University. (DL) This work was partially supported by NSF.

c 2001 American Mathematical Society 3347

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n m wandering vectors for U is denoted√ by W(U). If U = {D T : n, m ∈ Z} with (Tf)(t)=f(t − 1) and (Df)(t)= 2f(2t) for all f ∈ L2(R), then the complete wandering vectors are precisely the standard orthogonal (or orthonormal) wavelets, and more generally the complete wandering subspaces (rather, orthonormal bases for them) correspond to the orthogonal multi-wavelets. Suppose that x ∈W(U). The local commutant Cx(U)ofU at x, which is defined [DL] to be the set {A ∈ B(H):(AU − UA)x =0,U ∈U}, provides a good abstract machinery for creating new complete wandering vectors for U. Specifically, for every complete wandering vector y of U, there is a unique unitary operator V ∈Cx(U) such that y = Vxand vice versa (see Proposition 1.3 in [DL]). Another way to attempt to understand the wandering vector theory for a given unitary system U is to try to characterize all the unitary operators A in B(H) such that AW(U) ⊆W(U). A unitary operator with this property was called a wandering vector multiplier for U in [DL]. We use MU to denote the set of all such operators. It is obvious that MU is always a semi-group. A natural question is : Is MU a group? This seems hard to answer in general. A “good” characterization of the wandering vector multipliers for a specific unitary system U should be one which is strong enough to answer this basic test question. We conjecture that the answer is yes if U is a group. We will show that it is true for many groups. We will also show that it is true for irrational rotation unitary systems. It remains open for the orthogonal dyadic wavelet system, and this problem is one of the main motivations for this project, although there is much that has been accomplished here. Unitary groups with complete wandering vectors are exactly the left regular rep- resentation groups (under unitary equivalence, see the preliminaries section). A simple example is: Let H = L2(T), where T is the unit circle with normalized Lebesgue measure. Let U = Mz be the unitary operator corresponding to multi- plication by z. Then the constant function 1 is a complete wandering vector for U { n ∈ Z} ∈ 2 Z = Mz : n . An elementary argument shows that a function f L ( )is a complete wandering vector for U if and only if f is unimodular; i.e., |f(z)| =1, a.e. z ∈ T. Therefore the set of all the complete wandering vectors for U can be viewed as the set of all the unitary elements in the abelian von Neumann algebra L∞(T)viewedasvectorsinL2(T). (This idea can be generalized to the arbitrary discrete unitary group case.) Hence in this case the wandering vector multipliers are precisely those unitary operators on L2(T) which map unimodular functions to unimodular functions. Included are multiplication operators with unimodular symbols, and the composition operators with measure-preserving symbols. Even in this simple case the fact that MU is a group is perhaps not so obvious (see Example 2.10). It is true because an element of MU factors as a product of these two types (see Proposition 2.11 for the general abelian case), and the inverse of a composition operator with measure-preserving symbol is also a composition operator with mea- sure preserving symbol. The general (non-abelian) factorization result (see Remark 2.7) replaces the measure-preserving point mapping of the underlying measure space in the abelian case with an appropriate injective Jordan homomorphism of w∗(G) into itself which is isometric in the (noncommutative) Hilbert-Schmidt metric. For the unitary group case, the local commutant for a unitary group at a fixed complete wandering vector for the group is just the commutant of the group ([DL], Proposition 1.1). This suggests that there is a close relationship between wandering subspace theory and the von Neumann algebras generated by the left and the right regular representations of the group. The main result of this paper is in section 2 in

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which we obtain characterizations of wandering vector multipliers for some of the interesting groups including abelian and free groups. The methods also apply to the irrational rotation unitary systems that were studied in [Ha]. The results we get and the techniques involved show that wandering vector multipliers are interesting objects to study from the point of view, independent of possible applications. Given a complete wandering vector for a unitary group, we show that the di- agonal associated (resp. permutation) unitary operator (see definitions in section 3) are wandering vector multipliers if and only if they are induced by characters (resp. isomorphisms or anti-isomorphisms) of the group. The same results hold for the irrational rotation unitary systems. In particular, the last question in [Ha] which asks for a characterization of all the wandering vector multipliers for irra- tional rotation unitary systems is answered. The wandering vector multipliers for the wavelet system U = {DnT m : n, m ∈ Z} are called wavelet multipliers,and were first studied in [DL]. There is a special class of wavelet multipliers, namely functional wavelet multipliers, which play an essential role in understanding the phases of wavelets. These have been completely characterized (see [DGLL], [Wut]). This motivated us to try to characterize all the (unitary) wavelet multipliers, which is a problem posed in [DL]. While this general problem remains open, in section 5 we present several partial results on this.

1. Preliminaries

Two unitary systems U1 and U2 acting on Hilbert spaces H1 and H2, respectively, are said to be unitarily equivalent if there is a unitary operator T : H1 → H2 such U ∗ U W U W U ∗ that T 1T = 2.Inthiscase,T ( 1)= ( 2)andTMU1 T = MU2 .Given agroupG and two unitary representations π1 and π2,asusualwecallπ1 and π2 unitarily equivalent if there is a corresponding unitary operator T satisfying ∗ Tπ1(g)T = π2(g) for all g ∈G. We say a vector φ is a complete wandering vector for a unitary representation π of a group G if it is a complete wandering vector for the unitary system {π(g): g ∈G}. Let G be a discrete group with identity element e.Foreachg ∈G,lettg de- note the characteristic function at {g}.Then{tg : g ∈G}is an for l2(G). Define a lg (resp. rg), for each g,bylgth = tgh (resp. rgth = thg−1 ), then both lg and rg are unitary operators on l2(G). The unitary representation g → lg (resp. g → rg) is called the left (resp. right) regular repre- sentation of G.LetH be a Hilbert space. For any subset S of B(H), we will use w∗(S) to denote the von Neumann algebra generated by S and use S0 to denote the commutant of S; i.e., S0 = {T ∈ B(H):TS = ST,S ∈S}. The notations ∗ LG and RG are reserved to denote the von Neumann algebras w (lg : g ∈G)and ∗ w (rg : g ∈G), respectively. A group G is said to be an ICC group if the conjugacy class of each element other than the identity is infinite. It is a well-known fact that 0 LG = RG for all groups G, and also that both LG and RG are II1 factors when G is an ICC group (see [KR]). The following results will be used in section 2.

Proposition 1.1. Let G be a unitary group on a Hilbert space H which has a complete wandering subspace M.Thenw∗(G) is a finite von Neumann algebra. Moreover, G0 is finite if and only if dimM < ∞.

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Proof. Let {xi} be an orthonormal basis for M. By defining a unitary operator W : H → l2(G) ⊗ M such that

Wgxi = lgχe ⊗ xi, ∀i, ∀g ∈G,

we can assume that G has the form of {lg ⊗ I : g ∈G},whereI is the identity 0 operator on M. Since the commutant of {lg ⊗ I : g ∈G}is {lg : g ∈G} ⊗ B(M), 0 ∗ and {lg : g ∈G} and w (lg : g ∈G}are finite von Neumann algebras by [KR], the lemma follows.

Corollary 1.2. Let G be a unitary group which has a complete wandering subspace M.IfM has finite dimension n, then every n-dimensional wandering subspace N is complete. In general, if even dimM = ∞, every complete wandering subspace for G has the same dimension as M.

Proof. Let {xi : i =1,...,n} and {yi : i =1,...,n} be orthonormal bases for M and N, respectively. Define W ∈ B(H)by

Wgxi = gyi,i=1,...,n,g ∈G.

Then W is an isometry since {gxi : g ∈G,i=1,...,n} is an orthonormal basis for H and {gyi : g ∈G,i=1,...,n} is an orthonormal set. For any g,h ∈G,wehave 0 0 Wghxi = ghyi = gWhxi for i =1, 2,...,n. Hence W ∈G. By Proposition 1.1, G is a finite von Neumann algebra, which implies that W is a unitary operator. So N = WM must be a complete wandering subspace. For the second part, let n = dimM ≤∞and let N be a complete wandering subspace of dimension m for G. Suppose that n =6 m. We may assume that m

Let Gˆ be the dual group of an abelian group G, i.e., the compact group of all characters of G.Letθ : G→Gˆ be the duality correspondence between G and its ˆ second dual Gˆ, i.e., θ(g)=χg for any g ∈G,whereχg(ω)=ω(g),ω∈ Gˆ. The following fact is well-known in (cf. Prop. 2.2 in [BKM]) and will be used in section 2. Lemma 1.3. Let G be an abelian unitary group which has a cyclic unit vector f in H.ThenH is unitarily equivalent to the complex Hilbert space L2(Gˆ,σ),whereσ is a Borel probability measure on Gˆ. More specifically, there is a unitary operator W : H → L2(Gˆ,σ) such that (i) Wf =1,where1 denotes the function on Gˆ identically equal to 1; (ii) for ang g ∈G, the operator π(g)=WgW∗ is the multiplication operator (on 2 L (Gˆ,σ)) by the character χg:

π(g)ψ(ω)=χg(ω)ψ(ω) for all ψ ∈ L2(Gˆ,σ). Recall that a Jordan homomorphism from one C*-algebra into another C*- algebra is a linear mapping φ which satisfies φ(a∗)=(φ(a))∗ and φ(a2)=φ(a)2 for all a. The second condition is equivalent to the condition φ(ab + ba)=φ(a)φ(b)+ φ(b)φ(a) for all a, b.

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Lemma 1.4 ([BR], Proposition 3.2.2 or see [KR], 10.5.24). Let H be a Hilbert space, and suppose that φ is a Jordan homomorphism from a C∗-algebra A into B(H).LetB be the C∗-algebra generated by φ(A). Then there is a projection P in B0 ∩B00 such that a → φ(a)P is a *-homomorphism, and a → φ(a)(I − P ) is an *-antihomomorphism. Lemma 1.5 (Lemma 6 in [Ka]). Suppose that η is a Jordan homomorphism from a C*-algebra A into a C*-algebra B. Then for any element a, b ∈Aand any positive integer n, we have (i) η((ab)n +(ba)n)=(η(a)η(b))n +(η(b)η(a))n, (ii) η(aba)=η(a)η(b)η(a) We also need the following lemma with the proof due to Kadison (see the proof of Theorem 5 in [Ka]). Lemma 1.6. Let η be a Jordan homomorphism from a C*-algebra A into a C*- algebra B. Suppose that ||η|| ≤ 1 and η preserves the norm for self-adjoint elements. Then η is an isometry. Proof. Let a ∈Abe an arbitrary element of norm 1. Since both (aa∗)n and (a∗a)n are self-adjoint, we have 1=||a||4n = ||aa∗||2n = ||(aa∗)2n|| 1 = || ((aa∗)n + i(a∗a)n +((aa∗)n + i(a∗a)n)∗)|| · 2 1 || ((aa∗)n − i(a∗a)n +((aa∗)n − i(a∗a)n)∗)|| 2 ≤||(aa∗)n + i(a∗a)n|| · ||(aa∗)n − i(a∗a)n|| = ||(aa∗)2n +(a∗a)2n + i((a∗a)n(aa∗)n − (aa∗)n(a∗a)n)|| = ||η((aa∗)2n +(a∗a)2n)+iη((a∗a)n(aa∗)n − (aa∗)n(a∗a)n)|| = ||(η(a)η(a)∗)2n +(η(a∗)η(a))2n + i(η(a)∗η(a))n(η(a)η(a)∗)n − (η(a)η(a)∗)n(η(a)∗η(a))n)|| ≤ 4||η(a)||4n. for arbitrary positive integer n, where we use Lemma 1.5. Thus ||η(a)|| ≥ 1. Therefore ||η(a)|| =1sinceη is a contraction.

2. Wandering Vector Multipliers In this section, we will consider wandering vector multipliers for unitary groups. Since we will assume in this section that the unitary group G has a complete wan- dering vector, as we have seen in the proof of Proposition 1.1, it must be unitarily 2 equivalent to {lg : g ∈G}on L (G). So we can assume G has the form {lg : g ∈G}. Sometimes we do not distinguish lg from g when we consider g as a unitary operator. By Corollary 1.2, every wandering vector for G must be complete. Therefore a unit

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3352 DEGUANG HAN AND D. LARSON P { ∈G} vector η = g∈G λgtg is a complete wandering vector if and only if lhη : h is an orthonormal set, which, in turn, is equivalent to the condition  X 1,h= e, λ −1 λ = h g g 0,h=6 e, g∈G

where tg is the characteristic function of {g}.

Lemma 2.1. The two unitary groups {lg : g ∈G}and {rg : g ∈G}have the same complete wandering vectors.

Proof. Firstitisclearthatte is a complete wandering vector for both unitary groups. By Proposition 1.3 in [DL], η is a complete wandering vector for {rg : g ∈G} if and only if there is a unique unitary U in its commutant LG such that η = Ute. It is well known (see [KR]) that there is a conjugate linear unitary operator J on 2 ∗ l2(G) such that J = I,JAte = A te for all A ∈ LG and π : A → JAJ is a *- ∗ ∗ ∗ anti-isomorphism from LG onto RG .Thusη = Ute = JU te = JU Jte with JU J being a unitary in RU . Therefore, again by Proposition 1.3 in [DL], η is a complete wandering vector for {lg : g ∈G}. Similarly, every complete wandering vector for {lg : g ∈G}is also a complete wandering vector for {rg : g ∈G}. To characterize the wandering vector multipliers, it is convenient to first consider the abelian group case. As we mentioned in the introduction (also see Example { n ∈ Z} 2.10), the set of all the complete wandering vectors for Mz : n is precisely the set of all unimodular functions. Similarly if G is abelian, then, by Lemma G { ∈G} 2 Gˆ 1.3, we can assume = Mχg : g ,actingonL ( ,σ), with the complete wandering vector 1. By Corollary 1.2, every (unit) wandering vector is complete. It is easily seen that if |ψ(ω)| =1 a.e.,thenψ is a wandering vector. Conversely if ψ ∈ L2(Gˆ,σ) is a wandering (unit) vector, then Z 2 hπ(g)ψ,ψi = χg(ω)|ψ(ω)| dσ =0 Gˆ when g is not the identity of G. Since 1 is a complete wandering vector for π(G), the 2 2 set of vectors {χg}g∈G is an orthonormal basis for L (Gˆ,σ). It follows that |ψ| is a constant function, hence |ψ| is a constant. Since ψ is a unit vector, this constant is 1. Hence |ψ(ω)| = 1 a.e. So a unitary operator is a wandering vector multiplier if and only if |(Uf)(ω)| = 1 a.e. whenever |f(ω)| = 1 a.e. Thus the question of how to characterize all the wandering vector multipliers becomes the problem of characterizing all such operators U in a simple way. For the general (non-abelian) case, there is an analogue using some von Neumann algebra techniques. Let G be a unitary group acting on a Hilbert space H which has a complete wandering vector ψ and let M be the von Neumann algebra generated by G.Then M and its commutant M0 are finite von Neumann algebras which have a joint separating and cyclic trace vector ψ.Letτ be the faithful trace defined by τ(T )= hTψ,ψi.ThenM becomes a pre-Hilbert space with the inner product hS, T i = ∗ 2 τ(T S). Let L (M,τ) be the completion of M with respect to the norm kT k2 = (τ(T ∗T ))1/2.ThenL2(M,τ) is a Hilbert space. There is a natural way to represent G on L2(M,τ). Specifically, let π(g) ∈ B(L2(M,τ)) be defined by π(g)T = gT for 2 all T ∈M, and then extend it to L (M,τ) by using the inequality kgTk2 ≤kT k2. It follows that π(g) is a unitary and I is a complete wandering vector for π(G). Note that ψ is a separating vector for M. We can define W : Mψ →M⊂L2(M,τ)by

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WAψ = A.ThenW is isometric and can be extended to a unitary from H onto L2(M,τ). Also π(G) is unitarily equivalent to G via this unitary operator W .For this reason we will consider π(G) instead of G when necessary. Lemma 2.2. In the above notations, if x ∈ L2(M,τ) is a complete wandering vector for π(G),thenx is a unitary element in M, viewed as a vector of L2(M,τ). Proof. If x is a complete wandering vector for π(G), then W ∗x is a complete wan- dering vector for G. Thus, by Lemma 2.1, there exists a unitary operator U ∈ w∗(G) satisfying W ∗x = Uψ.Sox = WUψ. However, by the definition of W ,wehave WUψ = U. Hence x = U, as required. Proposition 2.3. With the notations above, an element x ∈ L2(M,τ) is a com- plete wandering vector for π(G) if and only if x is a unitary element in M. Proof. Let x be a unitary element in M.Then  0,g=6 I, hπ(g)x, xi = τ(x∗gx)=τ(gxx∗)=τ(g)= 1,g= I, where I is the identity operator in G.Thusx is a wandering vector for π(G)and hence complete by Corollary 1.2. The converse follows from Lemma 2.2. By Proposition 2.3, a wandering vector multiplier corresponds precisely to a unitary operator on L2(M,τ) that sends all the unitary elements in M to unitary elements in M. In the paper [Ka] concerning linear mappings from C*-algebras to C*-algebras, Kadison proved the following famous result: If φ is a linear isometry from a C*-algebra onto a C*-algebra, then φ is a Jordan isomorphism followed by left multiplication by a fixed unitary element, viz., φ(I). And a Jordan iso- morphism is, in turn by Lemma 1.4, the (direct) sum of a *-homomorphism and a *-antihomomorphism. We are dealing with unitary-preserving linear mappings here and we do not assume surjectivity. We will first prove that this kind of mapping is also a Jordan homomorphism followed by left multiplication by a fixed unitary element. The techniques used in [Ka] can be applied to this situation, yielding Proposition 2.5. It will, of course, follow from this that it is a linear isometry (Corollary 2.6) , but we do not see a way of directly proving it is an isometry and then obtain this decomposition using Kadison’s theorem. The elegant proof of the next lemma is taken from [Ka] and is included for completeness. Lemma 2.4. Let φ be a linear mapping from a C*-algebra A into a C*-algebra B ∗ ∗ such that kφk≤1 and φ(IA)=IB.Thenφ(a )=φ(a) for all a ∈A. Proof. First suppose that a is a self-adjoint element of norm 1 and let φ(a)=b + ic with b and c self-adjoint. If c =6 0, then there is a non-zero number β in the spectrum of c. We can assume that β>0. Then for n large enough, we have (1 + n2)1/2 <β+ n.Ifwenotethat kc + nIk≤kb + i(c + nI)k = kφ(a + inI)k≤ka + inIk, then we have ka + inIk =(1+n2)1/2 <β+ n ≤kc + nIk≤ka + inIk.

This is a contradiction. Hence c =0.Soφ(a) is self-adjoint. Now let a = a1 + ia2 ∗ ∗ be arbitrary with a1 and a2 self-adjoint. Then φ(a) = φ(a1) − iφ(a2)=φ(a ).

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The proof of the next proposition is an adaptation of the elegant argument in [Ka] to the unitary-preserving situation. Proposition 2.5. If φ is a unitary-preserving linear mapping from a C*-algebra A into a C*-algebra B such that φ(IA)=IB,thenφ is a Jordan homomorphism. Proof. First we prove that φ has norm one. In fact for any a ∈Awith kak≤1, 1 there are four unitary elements u1,u2,u3,u4 satisfying x = 4 (u1 + u2 + u3 + u4) (cf. Theorem 4.1.7 in [KR]). Thus kφ(a)k≤1sinceφ(u1),φ(u2),φ(u3),φ(u4)are unitaries. Hence kφk =1sinceφ(IA)=IB. Therefore, by Lemma 2.4, φ preserves adjoints. Moreover φ preserves order. In fact if a is a positive element with norm one, then φ(a) is self-adjoint and kφ(a) − Ik = kφ(a − I)k≤ka − Ik≤1. Thus φ(a) has non-negative spectrum. So it is positive. Now we verify that φ preserves the power structure. First let a be a self-adjoint element with norm less than or equal to one. Then u = a2 +i(I −a2)1/2 is a unitary element. Hence so is φ(u) by hypothesis. Thus we have I = φ(u)φ(u)∗ =(φ(a)+iφ((I − a2)1/2))(φ(a) − iφ((I − a2)1/2)) = φ(a)2 + φ((I − a2)1/2)2 + i(φ((I − a2)1/2)φ(a) − φ(a)φ((I − a2)1/2)). Therefore I = φ(a)2 +φ((I−a2)1/2)2, which implies that φ((I−a2)1/2)2 = I−φ(a)2. Since φ((I −a2)1/2) is a positive element, we obtain φ((I −a2)1/2)=(I −φ(a)2)1/2. For an arbitrary self-adjoint element a ∈A.Letα be non-zero real number such that αa has norm less than one. By using the binomial expressions: 1 1 (I − (αa)2)1/2 = I − α2a2 − α4a4 + ..., 2 8 1 1 (I − φ(αa)2)1/2 = I − α2φ(a)2 − α4φ(a)4 − ..., 2 8 and the relation φ((I − (αa)2)1/2)=(I − φ(αa)2)1/2, we have 1 1 1 1 φ(a2)+ α2φ(a4)+···= φ(a)2 + α2φ(a)4 + ... . 2 8 2 8 Thus it follows that φ(a2)=φ(a)2 by letting α tend to 0. For an arbitrary element a,writea = b + ic with b, c self-adjoint. Then φ(b + c)2 = φ((b + c)2),φ(b)2 = φ(b2)andφ(c)2 = φ(c2). Hence φ(bc + cb)=φ(b)φ(c)+φ(c)φ(b). Therefore φ(a2)=φ(b2 − c2 + i(bc + cb)) = φ(a)2 − φ(c)2 + i(φ(b)φ(c)+φ(c)φ(b)) = φ(a)2. The proof is complete. Corollary 2.6. If φ is a unitary-preserving linear mapping from one C*-algebra A into another C*-algebra B such that φ is one-to-one, then φ is an isometry.

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Proof. Let u = φ(I). Then u is a unitary element. By considering η = u−1φ,then η is a unitary-preserving mapping such that η(IA)=IB.Thusη is a one-to-one Jordan homomorphism by Proposition 2.5. Let a ∈Abe a self-adjoint element and let C be the C*-algebra generated by a and I.Sinceη preserves adjoints, and the power structure, and η is a contraction as proved in Proposition 2.5, η induces a one-to-one Jordan homomorphism from C into C. We claim that η preserves norm for self-adjoint elements. In fact, let a ∈Abe a self-adjoint operator and let b = ||η(a)||I + a.Thenη(b) ≥ 0. Note that b is self-adjoint. Let |b| be the positive square root of b.Thenη(|b|)2 = η(|b|2)=η(b2)=η(b)2.Sinceη(|b|) is positive (see Theorem 2.5), we have that η(|b|)=η(b) because of the uniqueness of positive square roots. Thus b = |b|, which implies that a ≥−||η(a)||I. Similarly, we have a ≤||η(a)||I. Therefore ||a|| ≤ ||η(a)||.Wealreadyknowthatη is a contraction. Hence η preserves norm for self-adjoint operators. Thus, by Lemma 1.6, we have that η is an isometry, and hence φ is an isometry. Remark 2.7. From Proposition 2.5 we have that any wandering vector multplier for ∗ π(G) can be factorized in the form MU CJ ,whereU is a unitary operator in w (G), and J is a Jordan homomorphism from w∗(G)intow∗(G) such that J(I)=I and with the additional property that J can be extended to a unitary operator CJ on L2(w∗(G),τ). In fact, let Φ ∈ B(L2(w∗(G),τ)) be a wandering vector multiplier for π(G). Then U := Φ(I) is a unitary by Proposition 2.3. Let CJ = MU ∗ Φandlet J be the restriction of C to w∗(G). Then C is also a wandering vector multiplier ∗ and CJ (I)=I.ThusJ is a unitary-preserving linear map from w (G)intoitself, and J(I)=I. Therefore, by Proposition 2.5, J is a Jordan homomorphism which satisfies the required requirements. The converse is also true. For the converse, by 10.5.22 in [KR], we have that J is unitary-preserving if J is a Jordan homomorphism 2 ∗ and J(I)=I.ThusMU CJ is a unitary operator on L (w (G),τ) which maps unitary elements in w∗(G) to unitary elements. Hence it is a wandering vector multiplier by Proposition 2.3. Lemma 2.8. Let φ be a Jordan homomorphism from a von Neumann algebra A into itself. If either A is abelian, or w∗(φ(A)) is a factor, then φ(A) is a *- subalgebra. Proof. Let B = w∗(φ(A)). If A is abelian, then φ is a *-homomorphism and hence φ(A) is clearly a *-subalgebra. In the general case, by Proposition 1.4, there is projection E in B∩B0 such that a → φ(a)E is a *-homomorphism and a → φ(a)E⊥ is a *-anti-homomorphism. Hence if B is a factor, then φ is either a homomorphism or an anti-homomorphism. In either case, we have that φ(A) is a *-subalgebra. Let M be a von Neumann algebra on a Hilbert space H. A closed operator A on H is said to be affiliated with M if M0D(A) ⊆ D(A)andAB ⊆ BA for all B ∈M0 (see [BR] or [KR]), where D(A) is the domain of A.Ifψ is a complete wandering vector for G,thenψ is a trace, cyclic and separating vector for both ∗ 0 M = w (G)andM . Thus we can define two conjugate linear operators S0 and F0 on Mψ by ∗ S0Aψ = A ψ for all A ∈Mand ∗ F0Bψ = B ψ

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for all B ∈M0. Using the properties of the trace vector, it follows that the closures S0 and F 0 are conjugate linear unitary operators. Theorem 2.9. If G is either finite or abelian or an ICC group acting on a Hilbert space H,andifG has a complete wandering vector, then MG is a group.

Proof. Let M, H, τ, ψ and π be as in Lemma 2.2. Let Φ be a unitary operator on 2 L (M,τ) such that Φ ∈ Mπ(G). In order to prove that Mπ(G) is a group, it suffices to show that Φ−1 is also a wandering vector multiplier. We will complete the proof by several steps. Since Φ is unitary-preserving, Φ(I) is a unitary in M. Replacing Φ by Φ(I)−1Φ, we can assume that Φ(I)=I. By Proposition 2.3, Proposition 2.5 and Corollary 2.6, we know that Φ is an isometry as well as a Jordan homomorphism from M into M. (1) Let N be the von Neumann algebra generated by Φ(M). We claim that N = M. In fact, by Proposition 2.3, Φ sends unitary operators in M to unitary operators in M.ThusN⊆Msince every element of M is a linear combination of four unitaries in M. Since Φ is a unitary operator on L2(M,τ)andM is τ- dense in L2(M,τ), we have that Φ(M)isτ-dense in L2(M,τ). This implies that Φ(M)ψ is dense in H.Thusψ is a cyclic and separating trace vector for N .(It is separating for N because N⊂Mand it is separating for M.)Hencewehave S0 and F0 related to N and ψ.LetA ∈M. By Proposition 2.5.9 in [BR], for Aψ ∈ D(S0)=H, there exists a closed operator Q such that Qψ = Aψ and Q is affiliated with N . Thus for any B ∈M0 ⊆N0,wehavethat

ABψ = BAψ = BQψ = QBψ.

Since ψ is cyclic for M0 and Q is closed, we obtain that Q is a and A = Q.SinceQ is a bounded operator affiliated with N ,wehavethatQ ∈N ⊆M. Thus Q = A since ψ is a separating vector for M. Hence A ∈N, and so we have N = M, as required. (2) We secondly claim that Φ(M)=M.LetA ∈Mwith norm one. By Lemma 2.8 and (1), Φ(M) is a *-subalgebra of M for the abelian and ICC group case and equal to M for the finite group case. Thus, by (1) in either case, we have that M is the strong closure of the *-subalgebra Φ(M). Using Kaplansky’s density theorem, there is a net {Aλ} in M such that Φ(Aλ) is in the unit ball of Φ(M)and Φ(Aλ)convergestoA in the strong operator topology. Thus {Aλ} is contained in the unit ball of M because Φ is an isometry for the norm topology of M.Since 2 Φ(Aλ)ψ → Aψ,wehavethat{Φ(Aλ)} is a Cauchy net in L (M,τ). Note that Φ 2 is viewed as a unitary operator on L (M,τ). Thus {Aλ} is also a Cauchy net in 2 L (M,τ), which implies that {Aλψ} converges. 0 We know that ψ is cyclic for M .Thus{Aλ} is a Cauchy net in the topology 0 of pointwise convergence on the dense subset M ψ since {Aλ} is a bounded net in norm. Therefore {Aλ} is a Cauchy net in the strong operator topology and thus converges to some operator B ∈M.SokΦ(Aλ)ψ − Φ(B)ψk = kΦ(Aλ − B)ψk = k(Aλ − B)ψk→0. Hence Aψ =Φ(B)ψ.Sinceψ separates M and A, Φ(B) ∈M, we have A =Φ(B). Therefore M =Φ(M), as required. (3) By (2) and Corollary 2.6, Φ is a surjective isometry on M.Sotheinverse of Φ is an isometry. A surjective isometry between unital C*-algebras preserves unitary elements [Ka]. Therefore, by Proposition 2.3, Φ−1 is a wandering vector multiplier. The proof is complete.

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Example 2.10. Now we consider the example mentioned in the introduction. Let 2 T U { n ∈ Z} H = L ( ), = Mz : n and let Mz be the unitary operator corresponding to multiplication by z. Then a function is a complete wandering vector for U if and only if it is unimodular. Any multiplication unitary operator Mf with f unimodular belongs to MU .Thereareothers.Letσ be a measure-preserving bijective mapping 2 −1 from T to itself. Define a unitary operator Aσ on L (T)by(Aσf)(z)=f(σ (z)) 2 for all f ∈ L (T). Then Aσ ∈ MU . Thus every element in the group generated by all the Mf and all the Aσ is a wandering vector multiplier. We claim that every wandering vector multiplier for U can be factored as the product of Aσ and Mf for some measure-preserving bijection mapping σ and some unimodular function f. ∗ ∞ In fact, suppose that T ∈ MU .Identifyw (U)withL (T). Then, by the proof of Theorem 2.9, T = V Φ, for some unitary operator V in w∗(U) (hence some unimodular function in L∞(T)) and some Φ : L∞(T) → L∞(T) such that Φ is a ∞ ∞ *-isomorphism of L (T)andkΦ(g)k2 = kgk2 for all g ∈ L (T). Then Φ induces a homomorphism of the Borel algebra (B, T), where B is the set of all Borel sets of T. Therefore there is a measurable one-to-one (modulo a null set) (see Theorem 4.7 on page 17 in [Pe]) mapping σ from T onto T such that Φ(f)(t)=f(σ(t)), a.e. ∞ on T. However the condition kΦ(g)k2 = kgk2 for all g ∈ L (T) implies that σ is measure-preserving. Hence we get the equality. This example can be extended to the arbitrary abelian group case in the following way.

Proposition 2.11. Let G be an abelian unitary group which has a complete wan- dering vector and let Gˆ, σ and π be as in Lemma 1.3. Then an operator T on L2(Gˆ,σ) is a wandering vector multiplier for π(G) if and only if T (f)(ω)= h(ω)f(Φ(ω)) for all f ∈L2(Gˆ,σ) for some unimodular function h and some measure- preserving mapping Φ from Gˆ onto Gˆ.

Proof. Since L∞(Gˆ,σ), the von Neumann algebra generated by π(G), is maximal abelian, it follows that it is unitarily equivalent to L∞(X, µ)onaHilbertspace L2(X, µ) for some compact set X in the complex plane and some finite Borel mea- sure µ (cf. Corollary 7.14 in [RR]). Thus, by Theorem 4.6 in [Pe, p.17], (B, Gˆ,σ)is Lebesgue in the sense of Definition 4.5 in [Pe, p.16], where B denotes the measure algebra. Now let T ∈ Mπ(G). Then, by the proof of Theorem 2.9, there is a unitary element h in w∗(π(G)) (=L∞(Gˆ,σ)) and an *-isomorphism α on w∗(π(G)) satisfying T (f)=hα(f) for all f ∈ L∞(Gˆ,σ)and

Z Z |α(f)(ω)|2dσ = |f(ω)|2dσ Gˆ Gˆ

for all f ∈ L2(Gˆ,σ). The *-isomorphism α induces an isomorphism from the measure algebra (B, Gˆ,σ) onto itself. Since (B, Gˆ,σ) is a Lebesgue space, by Theorem 4.7 in [Pe, p.17], there is a measurable one-to-one point mapping (modulo a measure zero set) Φ such that α(f)(ω)=f(Φ(ω)). By taking f to be a characteristic function on a set E ∈B,

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we have Z µ(Φ−1(E)) = |f(Φ(ω))|2dω ZGˆ = |α(f)(ω)|2dω ZGˆ = |f(ω)|2dω Gˆ = µ(E). Thus Φ is a measure-preserving mapping and the proof is complete.

Let I be the group of all surjective isometries on w∗(G). We have Theorem 2.12. Let G be an ICC unitary group acting on a Hilbert space H which has a complete wandering vector. Then MG is group isomorphic to I Proof. Let M = w∗(G). By the proof of Theorem 2.9, we only need to show that every Jordan isomorphism of M induces a wandering vector multiplier. Suppose that φ is a Jordan isomorphism on M. Then it must be either a *- automorphism or a *-anti-isomorphism since M is a factor. Assume that it is an automorphism. Then, by 9.6.27 in [KR], there is a unitary operator U acting on ∗ H such that φ(A)=UAU for each A ∈M. We claim that kφ(T )k2 = kT k2 for ∈M k k2 h i all T ,where T 2 = Tψ,Tψ for the given complete wandering vector ψ. ∗ Define τU (T )=hUTU ψ,ψi for each T ∈M.ThenτU is a normalized faithful normal trace of M. By the corollary of Theorem 3 in [Di, p.103], we have that τU = τ.Thuskφ(T )k2 = kT k2 for all T ∈M. Thisimpliesthatφ induces a unitary operator on H. Hence it induces a wandering vector multiplier since it preserves the unitary elements of M. If φ is a *-anti-isomorphism, the proof is similar. G G Corollary 2.13. Let 1 and 2 be ICC groups. If RG1 is *-isomorphic to RG2 ,

then MG1 is group isomorphic to MG2 . Let M be a von Neumann algebra acting on H.Anormalizer of M is a unitary ∗ operator A on H such that A MA = M.WeuseNG to denote the set of all ∗ normalizers for w (G). For convenience, we use NCG to denote the set of all unitary operators U with the property that U ∗w∗(G)U ⊆G0. Theorem 2.14. Let G be an ICC unitary group with a complete wandering vector. Then MG is equal to the group generated by NG ∪ NCG .

Proof. Let ψ ∈W(G) be arbitrary. We have known that τψ(T )=hTψ,ψi defines a faithful normalized trace on both w∗(G)andG0. ∗ Suppose that A ∈ NG. Define τAψ(T )=hA TAψ,ψi. By using the fact that ∗ ∗ ∗ ∗ A w (G)A = w (G), it follows that τAψ is also a faithful normalized trace for w (G). ∗ Since w (G)isafactor,wehavethatτψ = τAψ (cf. [Di]), which implies that Aψ is a wandering vector for G and hence complete by Corollary 1.2. Thus A ∈ MG. If A ∈ NCG. Define τAψ as above. Using the fact that ψ is also a faithful trace 0 ∗ 0 ∗ vector for G and that A TA∈G for all T ∈ w (G), it follows that τAψ is a faithful ∗ normalized trace for w (G). Thus τAψ = τ, which implies that Aψ ∈W(G). Hence A ∈ MG.

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From Theorem 2.9, MG is a group. Thus the group generated by NG and NCG is contained in MG. Conversely, let A ∈ MG. Fix a complete wandering vector ψ.ThenAψ ∈W(G). Thus there is a unitary operator U ∈ w∗(G) such that Aψ = Uψ.NotethatU is a normalizer for w∗(G). Replacing A by U −1A, we can assume that Aψ = ψ. By the proof of Theorem 2.9, there is a Jordan isomorphism Φ on w∗(G)such that AT ψ =Φ(T )ψ for all T ∈ w∗(G). Since w∗(G) is a factor, it follows that Φ is either a *-isomorphism or a *-anti-isomorphism. If Φ is a *-isomorphism, then for all S, T ∈ w∗(G)wehave AT Sψ =Φ(TS)ψ =Φ(T )Φ(S)ψ =Φ(T )ASψ. Thus AT =Φ(T )A for all T ∈ w∗(G)sinceψ is cyclic for w∗(G). Therefore ∗ ∗ ∗ ∗ AT A =Φ(T ) ∈ w (G). Since A ψ = ψ and A ∈ MG, by the same argument we also have A∗TA∈ w∗(G). Therefore A is a normalizer for w∗(G). If Φ is an *-anti-isomorphism, we need to show that A∗TA∈G0 for all T ∈ w∗(G). Let S, T, U, V ∈ w∗(G) be arbitrary. Then hUψ, A∗TASVψi = hAUψ, T ASV ψi = hΦ(U)ψ,TΦ(V )Φ(S)ψi = tr(Φ(S)∗Φ(V )∗T ∗Φ(U)) and hUψ, SA∗TAVψi = hAS∗Uψ, TAVψi = hΦ(U)Φ(S∗)ψ,TΦ(V )ψi = tr(Φ(V )∗T ∗Φ(U)Φ(S)∗). Since ψ is a trace vector for w∗(G), it follows that hUψ, A∗TASVψi = hUψ, SA∗TAVψi. Hence (A∗TA)S = S(A∗TA) for all S ∈ w∗(G)sinceψ is cyclic for w∗(G). Therefore A∗TA∈G0 for all T ∈ w∗(G), as required.

The proof of Theorem 2.14 also shows that if G is abelian, then MG ⊆ NG.The following example tells us that the equality is not necessarily true. Example 2.15. Let H = L2([0, 1]) and let

G = {Me2nπis : n ∈ Z}. Then, as we mentioned before, a function f ∈ H is a complete wandering vector for G if and only if |f(s)| =1,a.e. s ∈ [0, 1]. Let σ :[0, 1] → [0, 1] be a measurable bijection defined by  1 1  s, 0 ≤ s ≤ , σ(s)= 2 2 3 1 1 s − ,

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Let h ∈ L∞([0, 1]) such that   1 1  √ , 0 ≤ s ≤ , 2 2 h(s)= r  3 1  ,

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3. Diagonal and Permutation Wandering Vector Multipliers In this section we will study a special class of wandering vector multipliers which are either diagonal or permutation unitary operators induced by any fixed wander- ing vector for a unitary group G. It turns out that this class of multipliers can be completely characterized by either the dual group of G or the automorphism (resp. anti-automorphism) group of G. Forafunctionf : G→T (resp. a bijection σ of G), we can define a unitary operator Bf (resp. Aσ)byBf tg = f(g)tg (resp. Aσtg = tσ(g)) for all g ∈G.We will call f (resp. σ)awandering vector multiplier function (resp. wandering vector multiplier mapping)onG if Bf (resp. Aσ) is a wandering vector multiplier for G. Using Proposition 2.5, we have the following characterization of wandering vector multiplier functions and mappings. Theorem 3.1. (i) f is a wandering vector multiplier function if and only if there exists a modulus one complex number λ and a character ω of G such that f(g)= λω(g) for all g ∈G. (ii) σ is a wandering vector multiplier mapping of G if and only if there exists an element h ∈G and an automorphism or anti-automorphism τ such that σ(g)= hτ(g) for all g ∈G. Proof. (i) For necessity, by considering f(e)−1f, we can assume that f(e) = 1. Sup- ∗ pose that Bf ∈ MG. By Proposition 2.5, Bf is a Jordan homomorphism on w (G). Thus, for any g,h ∈G,wehavethatAf (gh + hg)=Af (g)Af (h)+Af (h)Af (g). That is f(gh)gh + f(hg)hg = f(g)f(h)gh + f(h)f(g)hg.Ifgh = hg,weobtain f(gh)=f(g)f(h). Suppose that gh =6 hg,thenby

h(f(gh)gh + f(hg)hg)te,tghi = h(f(g)f(h)gh + f(h)f(g)hg)te,tghi, we have

hf(gh)tgh + f(hg)thg,tghi = hf(g)f(h)tgh,tghi + hf(h)f(g)thg,tghi.

Note that htgh,thgi =0andhtgh,tghi =1.Wegetf(gh)=f(g)f(h). Thus f(gh)=f(g)f(h) for all g,hP ∈G. Therefore f is a character of G. { ∈ For sufficiency, let η = g∈G λgtg be a complete wandering vector for lg : g G}. Then, for any h ∈G,wehave X X hlhBωη, Bωηi = h λgω(g)thg, λgω(g)tgi ∈G ∈G gX g X −1 = h λh−1gω(h g)tg, λgω(g)tgi ∈G ∈G Xg g −1 = λh−1gω(h g)λgω(g) ∈ gX[G −1 = ω(h )λh−1gλg g∈G = ω(h−1)hl η, ηi ( h 1,h= e, = 0,h=6 e.

Thus Bωη is a wandering vector which also complete by Lemma 1.2. Hence Bω is a wandering vector multiplier.

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(ii) For necessity we can also assume that σ(e)=e.IfAσ is a wandering vector multiplier, then Aσ induces a Jordan homomorphism on LG.Thusσ(gh)+σ(hg)= σ(g)σ(h)+σ(h)σ(g). So

htσ(gh) + tσ(hg),tσ(gh)i = htσ(g)σ(h) + tσhσ(g),tσ(gh)i.

This equality, together with hta,tbi = δa,b for all a, b ∈G, implies that either σ(gh)=σ(g)σ(h)orσ(gh)=σ(h)σ(g). Thus, by a result due to L. K. Hua (see Lemma 1 in [JR]), we know that σ is either an automorphism or an anti- automorphism. P Conversely, we assume that σ is an automorphism.P Let η = g∈G λgtg be a ∈G complete wandering vector. Then Aση = g∈G λgtσ(g).Thus,foreachh ,we have X X hlhAση, Aσηi = h λgthσ(g), λgtσ(g)i ∈G ∈G gX g X = h λσ−1(h−1g)tg, λσ−1(g)tgi ∈G ∈G Xg g = λσ−1(h−1)σ−1(g)λσ−1(g) g∈G

= hlσ(h−1)η, ηi.

Thus {lhAση} is an orthonormal set, which implies that Aση is a complete wandering vector by Lemma 1.2. If σ is an anti-automorphsim, we use Lemma 2.1, and the proof is similar.

Let U be a unitary system which has a complete wandering vector. If U has the the property that U = UU0 with U0 beingasubsetofU which is a group, it is 0 ∗ known that the product of a unitary in U and a unitary in w (U0) is a wandering ∗ vector multiplier (see [DL]), where w (U0) is the von Neumann algebra generated by U0. Dai and Larson asked in [DL] whether the converse is true. This was called the factorization problem. There are some counterexamples for the non-group case (see [LMT], [Ha]). The following proposition provides some counterexamples for the group case (i.e. the case where U itself is a group). The following proposition tells us that many Aσ, Bf ’s do not have the factoriza- tion property.

Proposition 3.2. Let U = {lg : g ∈G}and let f : G→C (resp. σ : G→G)bea unimodular function (resp. bijection). 0 ∗ (i) If G is an abelian group, then Aσ (resp. Bf ) belongs to U w (U) if and only if σ = id (resp. f(g)=1for all g ∈G). (ii) If σ is either (a) an antimorphism and {σ(h−1)h : h ∈G}is an infinite set, or (b) a non-identity automorphism and {σ(h−1)gh : h ∈G}is an infinite set for 0 ∗ all but finite number of g ∈G ,thenAσ does not belong to U w (U),

Proof. (i) Suppose that Aσ has the factorization property. Then it must belong to 0 LG (= LG)sinceLG is an abelian von Neumann algebra with a cyclic vector which implies that it is a maximal von Neumann algebra. Hence

Aσlg = lgAσ

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for all g ∈G. Apply te to the above equality, we get tσ(g) = Aσtg = Aσlgte = lgAσte = lgtσ(e) = tg for each g ∈G. Hence σ(g)=g for all g ∈G, which implies that σ = id. Similarly, suppose Bf has the factorization property. Then

Bf lg = lgBf

for all g ∈G, which implies that (by applying te to the equality) f(g)tg = f(e)tg. Hence f(g)=f(e)=1. (ii) Assume (a) holds. If Aσ has the factorization property, then there is a ∈ unitary operator UPin LG such that AσU RG .ThusAσUlg =PlgAσU for all ∈G − − Pg .LetUte = h∈G λhth.ThenUtg = Urg 1 te = rg 1 Ute = h∈G λhthg = ∈G λhg−1 th.So h X X AσUlgte = AσUtg = λhg−1 tσh = λσ−1 (h)g−1 th, h∈G h∈G and X X −1 −1 lgAσUte = λhtgσh = λσ (g h)th. h∈G h∈G Thus we get

λσ−1(h)g−1 = λσ−1 (g−1h) for all g,h ∈G.Writeσ−1(g−1h)=k.wegeth = gσ(k). So σ−1(h)g−1 = − − 1 1 − − ∈G kσ (g)g . This implies that λk = λkσ 1 (g)g 1 holds for all Pk, g .Since { −1 −1 ∈G} | |2 ∞ σ (g)g ; g is an infinite set, we have λk = 0 because h∈G lh < , which implies that Ute = 0, a contradiction. Hence Aσ does not have the factor- ization property. The proof is similar for the case (b). 4. Irrational Rotation Unitary Systems In [Ha], the first named author of this paper studied wandering vectors of irra- tional rotation unitary systems Uθ. These are not unitary groups. Some results concerning wandering subspaces and wandering vector multipliers for irrational ro- tation unitary systems were obtained in [Ha]. In this section, we point out that most of the results in section 2 remain true for irrational rotation unitary systems. In particular we answer the last question raised in [Ha]. Let θ be an irrational number in (0, 1). Recall that ([Ha]) an irrational rotation unitary system U is a system of the form {U nV m : n, m ∈ Z} with UV = e2πiθVU. If ψ is a complete wandering vector for U, then, by Theorem 1 and Theorem 12 in 0 ∗ 0 [Ha], we know that Cψ(U)=U , ψ is a faithful trace vector for w (G)andU ,and w∗(U) is a finite factor. These properties imply that Lemma 2.1, 2.2, 2.3, Theorem 2.9, Theorem 2.12 and Theorem 2.14 remain true for irrational rotation unitary systems U. We leave the routine checking of this to the reader. Thus the following answers the last problem posted in [Ha], which asks for a complete characterization of wandering vector multipliers for irrational rotation unitary systems. Theorem 4.1. Suppose that U is an irrational rotation unitary system such that W(U) is non-empty. Then MU is a group. Moreover, it is equal to the group generated by all the normalizers of w∗(U) and all the unitaries A with the property A∗w∗(U)A ⊆U0.

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Similar to the group case, we also have a complete characterization for the di- agonal wandering vector multipliers with respect to a fixed complete wandering vector. Let f : Z ⊗ Z → T be a function and let Bf be a unitary defined by n m n m n m Bf U V ψ = f(n, m)U V ψ on the orthonormal basis {U V ψ : n, m ∈ Z}. Then we have

Proposition 4.2. The unitary operator Bf is a wandering vector multiplier if and only if there exist two characters ω and σ of Z such that f(n, m)/f(0, 0) = ω(n)σ(m) for all n, m ∈ Z. Proof. Sufficiency is contained in Theorem 10 in [Ha]. So we only need to prove the necessity. Suppose that Bf ∈ MG. Clearly we can assume that f(0, 0) = 1. By Proposition 2.3 and Proposition 2.5 for the irrational rotation unitary system case, ∗ Bf induces a Jordan homomorphism on w (U). Thus it must be an automorphism or anti-automorphism since w∗(U)isafactor.Thisimpliesthatω(n):=f(n, 0) and σ(n):=f(0,n) define two characters of Z.From n m n m n m n m f(n, m)U V = Bf (U V )=Bf U Bf V = f(n, 0)f(0,m)U V , we get that F (n, m)=ω(n)σ(m) for all n, m ∈ Z. Thus the proof is complete. Let α and τ be two bijections on Z. Fix a complete wandering vector ψ ∈W(U). Define a unitary operator Aα,τ by n m α(n) τ(m) Aα,τ U V ψ = U V ψ for all n, m ∈ Z.Wehave

Proposition 4.3. Suppose that α(0) = τ(0) = 0.ThenAα,τ is a wandering vector multiplier if and only if both α and τ are group isomorphisms of Z.

Proof. For convenience let A = Aα,τ . First assume that A is a wandering vector multiplier. By Proposition 2.3 and Proposition 2.5 (for the irrational rotation unitary system case), A induces a Jordan homomorphism Φ on w∗(U) such that AT ψ =Φ(T )ψ for all T ∈ w∗(U). Since w∗(U) is a factor, it follows from Lemma 2.8 that Φ is either a *-homomorphism or a *-antihomomorphism. If Φ is a *-homomorphism, then U α(n+m) = AU n+mψ =Φ(U n+m)ψ =Φ(U n)Φ(U m)ψ for all n, m ∈ Z.ThusU α(n+m) =Φ(U n)Φ(U m)sinceψ separates w∗(G). In particular U α(n) =Φ(U n) for all n ∈ Z.ThusU α(n+m) = U α(n)U α(m),which implies that α(n+m)=α(n)+α(m) for all n, m ∈ Z. Hence α is an automorphism of Z. If Φ is an *-antihomomorphism, the same argument works. Similarly, τ is an automorphism of Z. The converse is trivial since there are only two automorphisms of Z,theidentity and the inverse automorphism: n →−n. 5. Wavelet Multipliers For a general unitary system it is not clear how to characterize the set of all the wandering vector multipliers. In this section we investigate some special wandering vector multipliers for the unitary systems which are the ordered product of two unitary groups. This is an interesting class which contains the wavelet systems in

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L2(Rn). When we consider wavelet systems, for simplicity in this paper we will only consider the one-dimensional dyadic wavelet system. The Fourier transform, fˆ, of a function f ∈ L1(R) ∩ L2(R) is defined by Z 1 fˆ(ξ)=√ f(s)e−isξds, 2π R where ds means integration with respect to Lebesgue measure. This transformation can be uniquely extended to an unitary operator F on L2(R). We write Dˆ =: −1 ˆ −1 ˆ −1 ˆ FDF and T =: FT F . It is not hard to check that D = D and T = Me−is , −is where Me−is is the multiplication unitary operator with symbol e . The unitary { ˆ m ˆn ∈ Z} U system D T : n, m is denoted by D,ˆ Tˆ. As in [DL], the wandering vector multipliers for UD,T will be called wavelet multipliers. For a unitary system U = U0U1 which is the ordered product of two unitary groups U0 and U1, we always assume in this section that U0 ∩U1 = {I} and that W(U)isnon-empty.

Lemma 5.1. Let U = U0U1 be a unitary system on a Hilbert space H of the above ∗ U ∗ ∗ ∈ ∗ U ∈U form. If A normalizes w ( 1) and U0AU0 A w ( 1) for all U0 0,thenAψ is acyclicvectorforU whenever ψ is. ∗ Proof. Note that A also satisfies the above condition. Thus for any U0 ∈U0,we have ∗ ∗ ∗ U0w (U1)Aψ = U0A(A w (U1)A)ψ ∗ = U0Aw (U1)ψ ∗ ∗ ∗ U = AU0(U0 A U0A)w ( 1)ψ ∗ = AU0w (U1)ψ. Hence Aψ is cyclic for U if ψ is. Proposition 5.2. Let U be as in Lemma 5.1. Suppose that A ∈ B(H) is a unitary operator satisfying ∈U0 (i) A 1, ∗ ∗ ∈ ∗ U ∈U (ii) U0AU0 A w ( ) for all U0 0. Then A is a wandering vector multiplier for U.

Proof. By Lemma 5.1, it suffices to check the orthonormality of {U0U1Aψ : U0 ∈ U0,U1 ∈U1}. If U0 = I,then{U1Aψ : U1 ∈U1} is clearly an orthonormal set since U1A = AU1 and {U1ψ : U1 ∈U1} is an orthonormal set. Now let U0 ∈U0 such that U0 =6 I. ∗ ∈ ∗ U Write U0AU0 = BA = AB for some unitary operator B w ( 1). Then for all U, V ∈U1 we have

hU0UAψ, Vψi = hU0AUψ, AV ψi h ∗ i = U0AU0 U0Uψ, AVψ

= hABU0Uψ, AVψi

= hBU0Uψ, Vψi ∗ = hU0Uψ, B Vψi. ∗ ∗ ∗ Note that B ∈ w (U1). Thus hU0UAψ, Vψi = hU0Uψ, B Vψi = 0, as required.

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If f is a unimodular function, then it was proved in [DGLL] and [Wut], by using the fundamental equation characterizations of wavelets, that Mf is a wavelet multiplier if and only if f(2s)/f(s)is2π-periodic (a.e. s ∈ R). Proposition 5.2 does not imply this since the 2π-periodic property of f(2s)/f(s) can only guarantee that ˆ −n ˆ n −1 ∈ ∗ ˆ ≥ D Mf D Mf w (T ) for all n 0. For some special unitary group U1, Proposition 5.2 can be extended as follows:

Theorem 5.3. Let U be as in Lemma 5.1 such that U1 is an ICC group. Suppose that A ∈ B(H) is a unitary operator such that ∗ (i) A normalizes w (U1),and ∗ ∗ ∈ ∗ U ∈U (ii) U0AU0 A w ( 1) for all U0 0. Then A is a wandering vector multiplier for U. Proof. Let ψ ∈W(U) be arbitrary. By Lemma 5.1 and the proof of Proposition 5.2, it suffices to prove the orthonormality of the set {UAψ : U ∈U1}.Let ∗ Hψ =[w (U1)ψ], the closed subspace generated by U1ψ.ThenHψ is an invariant subspace of U1.LetG be the restriction unitary group of U1 to Hψ.Fromthe orthonormality of {U1ψ : U1 ∈U1},wehavethatG is also an ICC group which is group isomorphic to U1.Alsoobservethatψ is a complete wandering vector for G. Thus M, the von Neumann algebra generated by G,isafactor. Define a linear mapping Ψ : M→Mby ∗ | Φ(m)=A mA Hψ . ∗ ∗ Note that A mA ∈ w (U1). Then Φ is a *-isomorphism. Since ψ is both cyclic and separating for M, it follows from Theorem 7.2.9 in [KR] that there is a unitary ∗ operator B ∈ B(Hψ) such that Φ(m)=B mB. Thus B is a normalizer of M.By Theorem 2.14, B is a wandering vector multiplier for G.SoBψ ∈W(G). Therefore for any U ∈U1 such that U =6 I,weobtain hUAψ, Aψi = hA∗UAψ, ψi = hB∗UBψ, ψi = hUBψ, Bψi =0, as required.

If we consider the special case that U0 is the trivial group (i.e., U0 = {I}), then Example 2.15 tells us that Theorem 5.3 is not valid in general even when G1 is abelian. However in the rest of this section we will show that Theorem 5.3 is true U for wavelet systems Dˆ ,Tˆ. Lemma 5.4. Let φ : R → R be a measurable bijection such that (i) φ(2s)=2φ(s),a.e.s∈ R,and −1 (ii) both einφ(s) and einφ (s) are 2π-periodic almost everywhere for all n ∈ Z. Then φ is measure preserving. Proof. Let µ denote Lebesgue measure on R. We first claim that µ(φ(0, 2π]) = 2π. For every n ∈ Z, let

Fn = {s ∈ (0, 2π]: φ(s) − 2nπ ∈ (0, 2π]}.

Then {Fn : n ∈ Z} forms a partition of (0, 2π]. In fact, if s ∈ Fn ∩ Fm and n =6 m, ∈ ∩ { ∈ Z} then φ(s) ((0, 2π]+2nπ) ((0, 2π]+2mπ), which isS impossible. Thus Fn : n ∈ R is a disjoint family. Also for any s (0, 2π], since k∈Z((0, 2π]+2kπ)= ,there exists n such that φ(s) − 2nπ ∈ (0, 2π]. Hence s ∈ Fn.

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We check that {φ(Fn) − 2nπ : n ∈ Z} also forms a partition of (0, 2π]. Suppose that φ(s) − 2nπ = φ(t) − 2mπ for some n =6 m and for some s ∈ Fn and t ∈ Fm. Then φ−1(ψ(s) − 2nπ)=φ−1(φ(t) − 2mπ). So, by the assumption (ii), there is l ∈ Z such that s − t =2lπ. This implies l =0 since s, t ∈ (0, 2π], which is a contradiction. Thus {φ(Fn)+2nπ : n ∈ Z} is a disjoint family. To show that the union is (0, 2π], let t ∈ (0, 2π]. Then there is r such that φ(r)=t.Writer = s +2mπ with s ∈ (0, 2π]. Then by (ii)wehave t = φ(r)=φ(s +2mπ)=φ(s) − 2lπ

for some l ∈ Z. Hence s ∈ Fl and so t ∈ φ(Fl) − 2lπ, as required. Now we have [ [ µ(φ(0, 2π]) = µ(φ( Fn)) = µ( φ(Fn)) ∈Z ∈Z X n Xn = µ(φ(Fn)) = µ(φ(Fn) − 2nπ) ∈Z ∈Z n [ n = µ( (φ(Fn) − 2nπ)) = µ((0, 2π]) = 2π. n∈Z Next we claim that µ(φ(E + kπ)) = µ(φ(E)) for any measurable set E and any k ∈ Z. For any s ∈ E, conditions (i) and (ii) imply that there is l ∈ Z such that 2φ(s + kπ)=φ(2s +2kπ)=φ(2s)+2lπ =2φ(s)+2lπ. Hence φ(s + kπ)=φ(s)+lπ. Let E = {s ∈ E : φ(s + kπ)=φ(s)+lπ}.Then{E } is a partition of E.Thus l [ l µ(φ(E + kπ)) = µ( φ(El + kπ)) ∈Z [l = µ(φ(El + kπ)) ∈Z l[ = µ(φ(El)) ∈Z l [ = µ(φ( El)) = µ(φ(E)). l∈Z Finally, we claim that φ is measure preserving. For any k ∈ Z, by the above two claims, we have µ(φ((kπ, (k +1)π])) = µ(φ((0,π]+kπ)) = µ(φ((0,π])) 1 1 = µ( φ((0, 2π])) = µ(φ((0, 2π])) = π. 2 2 Thus k k +1 1 π µ(φ(( π, π])) = µ( φ((kπ, (k +1)π])) = 2n 2n 2n 2n for all n, k ∈ Z.Notethat k k +1 {( π, π]: k, n ∈ Z} 2n 2n generates the Borel structure of R.Thusφ is measure preserving.

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We say that two measurable sets E and F of R are translation congruent modulo 2π if there exists a measurable bijection σ : E → F such that σ(s)− s is an integral multiple of 2π for each s ∈ E. Lemma 5.5. Suppose that φ is as in Lemma 5.4. Then φ((0, 2π]) is 2π-translation congruent to (0, 2π]. Proof. Let t ∈ (0, 2π]andletφ(r)=t.Writer = s +2mπ for some (unique) s ∈ (0, 2π] and some (unique) m ∈ Z.Sinceeiφ(s) is 2π-periodic, there is k ∈ Z such that t = φ(s +2mπ)=φ(s)+2kπ.Sot − φ(s) ∈ 2Zπ.Sinceφ is measure- preserving, we get that φ((0, 2π]) is 2π-translation congruent to (0, 2π].

Lemma 5.6. Let A be a unitary operator such that (i) A normalizes w∗(Tˆ), (ii) Dˆ nADˆ −nA−1 ∈ w∗(Tˆ). Then there exists a measurable bijection φ : R → R and a function f ∈ L∞(R) −1 2 such that A = BφMf ,where(Bφh)(s)=h(φ (s)) for all h ∈ L (R). n m −n m Proof. Since (D T D )h(s)=h(s + 2n ) and the set of all the dyadic numbers { m ∈ Z} R { n m −n ∈ Z} 2n : n, m is dense in , it follows that D T D : n, m generates a maximal abelian von Neumann algebra. Thus the von Neumann algebra M −n n ∞ generated by {D Meims D : m, n ∈ Z} is {Mg : g ∈ L (R)}. −1 −n −n −1 For any n ∈ Z,writeA D = D A Mg for some 2π-periodic unimodular function g.Then −1 −n ∗ ˆ n −n −1 ∗ ˆ ∗ n A (D w (T )D )A = D A Mgw (T )Mg AD = D−nA−1w∗(Tˆ)ADn = D−nw∗(Tˆ)Dn. Thus A normalizes M. Define an isomorphism Φ on M by ∗ ∞ Φ(Mg)=A MgA, g ∈ L (R). Then, by Theorem 4.7 in [Pe, p. 16], there is a measurable bijective mapping φ : R → R ∈ ∞ R −1 −1 such that Φ(Mg)=Mg(φ(s)) for all g L ( ). That is, A MgA = Bφ MgBφ. −1 M0 M M M0 Therefore Bφ A is in the commutant of .Since = , it follows that ∞ there is a function f ∈ L (R) such that A = BφMf , as required.

Theorem 5.7. Let A be as in Lemma 5.5. Then F −1AF is a wavelet multiplier.

−1 −1 ∗ Proof. Let A = BφMf be as in Lemma 5.5. From DAD A ∈ w (Tˆ), there is a 2π-periodic unimodular function k(s) such that (DAD−1A−1g)(s)=k(s)g(s) for all g ∈ L2(R). Equivalently, f(φ−1(2s)) φ−1(2s) g(φ( )) = k(s)g(s) φ−1(2s) 2 f( 2 ) for all g ∈ L2(R). Let E be an arbitrary set of finite measure and let φ−1(2E) F = E ∪ φ( ). 2

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Applying g = χF to the above equality, we get f(φ−1(2s)) k(s)= ,s∈ E. φ−1(2s) f( 2 ) Since E is arbitrary we have the equality for all S ∈ R.Thus φ−1(2s) h(φ( )) = h(s) 2 2 0 for h ∈ L (R). Therefore φ(2s)=2φ(s), and thus Bφ ∈{D} . ∗ ˆ ∈ ∗ ˆ 0 −1 ∗ ˆ Since A normalizes w (T )andMf w (T ) ,wehavethatBψ w (T )Bφ = −1 w∗(Tˆ). This implies that both eilφ(s) and eilφ (s) are 2π-periodic. Therefore, by Lemma 5.5, φ is measure preserving, and hence both Bφ and Mf are unitary. We −1 −1 need to show that both F BφF and F Mf F are wavelet multipliers. Since n n −1 −1 −n n −n −1 −1 ∈ ∗ ˆ D BφMf D Mf Bφ D = Bφ(D Mf D Mf )Bφ w (T ) −1 ∗ ˆ ∗ ˆ n −n −1 ∈ ∗ ˆ and Bφ w (T )Bφ = w (T ), it follows that D Mf D Mf w (T ). Thus, by −1 Corollary 5.3 (or Proposition 5.2), F Mf F is a wavelet multiplier. Observe that we already know that Bφ also satisfies the conditions (i)and(ii). −1 To show that F BφF is a wavelet multiplier, by the proof of Theorem 5.3, it { ˆl ˆ ∈ Z} suffices toP show that T Bφh : l is an orthonormal set for any wavelet h. | |2 ∈ R Note that k∈Z h(s +2kπ) =1,a.e.s . Using Lemma 5.4, Lemma 5.5 and the fact that eilφ(s) is 2π-periodic, we have Z l ils −1 2 hTˆ Bφh,ˆ Bφhˆi = e |h(φ (s))| ds ZR = eilφ(s)|h(s)|2ds R Z X 2(k+1)π = eilφ(s)|h(s)|2ds ∈Z 2kπ Zk 2π X = eilφ(s) |h(s +2kπ)|2ds 0 ∈Z Z k Z 2π 2π = eilφ(s) = eilsds. 0 0 l Thus {Tˆ Bφh : l ∈ Z} is an orthonormal set.

Questions. Does every wavelet multiplier A have the form Aˆ = BσMf for some function f ∈ L∞(R) and some measurable bijection σ : R → R?Aretheonly −1 measurable bijections σ for which F BσF is a wavelet multiplier the identity bijection s → s and the inverse bijection s →−s? Another Direction. The wandering vector multipliers for a unitary system U that were investigated in [DL], [Ha] and in the present article are by definition uni- tary operators on the underlying Hilbert space that map complete wandering vectors for U to complete wandering vectors for U. However, as pointed out in Remark 2.17 (ii), a bounded invertible operator which maps complete wandering vectors to complete wandering vectors need not be unitary. It might be worthwhile to try to

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extend known results for unitary wandering vector multipliers to wandering vector multipliers which are not necessarily unitary whenever possible. For instance, in Example 2.10, let σ be a measurable bijection of T which is not measure-preserving but is such that the composition operator Aσ is bounded and invertible. Then Aσ maps unimodular functions to unimodular functions on T, so is a wandering vector multiplier which is essentially different from the type we have considered in this article. In the general group unitary system case the same type of non-unitary wandering vector multiplier can arise from a Jordan isomorphism of w∗(G)which is not isometric in the Hilbert-Schmidt metric on w∗(G) but is bounded above and below so extends to a (non-unitary) bounded invertible operator on L2(w∗(G),τ). In fact, as pointed out in Remark 2.17 (ii), there is a general factorization result for this case. In [Wut] it was not taken as a matter of definition that wavelet multiplier functions were unimodular, but rather, it was proven that a wavelet multiplier func- tion must be unimodular. This is in the same spirit. (Guido Weiss was the member of the WUTAM CONSORTIUM who suggested that the unimodularity should be proven and not simply assumed.) The following interesting open question remains: Question. Is an (operator) wavelet multiplier which is a bounded linear operator necessarily a unitary operator? References

[BKM]V.Bergelson,I.Kornfeld,andB.Mityagin,Weakly wandering vectors for unitary action of commutative groups, J. Funct. Anal., 126 (1994), 274-304. MR 96c:47008 [BR] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics I, Springer-Verlag, 1987. MR 98e:82004 [DL] X. Dai and D. Larson, Wandering vectors for unitary systems and orthogonal wavelets, Memoirs Amer. Math. Soc., 134 (1998), no. 640. MR 98m:47067 [DGLL] X. Dai, Q, Gu, D. Larson and R. Liang, Wavelet multipliers and the phase of an or- thonormal wavelet, preprint. [Di] J. Diximer, Von Neumann Algebras, North-Holland, 1981. MR 83a:46004 [Ha] D. Han, Wandering vectors for irrational rotation unitary systems, Trans. Amer. Math. Soc., 350 (1998), 321-329. MR 98k:47089 [JR] N. Jacobson and C. Rickart, Jordan homomorphisms of rings, Tran.Amer.Math.Soc.,69 (1950), 479-502. MR 12:387h [Ka] R. Kadison, Isometries of operator algebras, Ann. of Math., 54 (1951), 325-338. MR 13:256a [KR] R. Kadison and J. Ringrose, “Fundamentals of the Theory of Operator Algebras,” vols. I and II, Academic Press, Orlando, 1983 and 1986. MR 85j:46099; MR 88d:46106 [LMT] W. Li, J. McCarthy and D. Timotin, A note on wavelets for unitary systems, unpublished note. [Pe] K. Petersen, Ergodic Theory, Cambridge University Press, 1995. MR 95k:28003 [RR] H. Radjavi and P. Rosenthal, Invariant Subspaces, Springer-Velag, 1973. MR 51:3924 [Wut] The Wutam Consortium, Basic properties of wavelets, J. Fourier Anal. Appl., 4 (1998), 575–594. MR 99i:42056

Department of Mathematics, University of Central Florida, Orlando, Florida 32816- 1364 E-mail address: [email protected] Department of Mathematics, Texas A&M University, College Station, Texas 77843 E-mail address: [email protected]

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